### 4.1. The Riemann-Liouville Derivative Case

We will consider a nonautonomous fractional differential system with Riemann-Liouville derivative

under the initial conditions

where , matrix , is a continuous matrix, and . The main results of this subsection are derived as follows.

Theorem 4.1.

If the matrix such that , the critical eigenvalues which satisfy have the same algebraic and geometric multiplicities, and is bounded, then the zero solution of (4.1) is stable.

Proof.

Applying the Laplace transform, we can get the solution of (4.1)-(4.2),

From the proof of Theorem 3.1, the matrix is bounded for . Therefore, there exist positive numbers , such that (). Now, we can get the estimate of solution

Applying the Gronwall inequality (2.17) leads to

Thus, we derive that is bounded according to the condition , that is, the zero solution of (4.1) is stable. The proof is completed.

Similarly, we can derive the following conclusion.

Theorem 4.2.

If the matrix such that , , and (, ) for , then the zero solution of (4.1) is asymptotically stable.

Proof.

From the proof of Theorem 3.1, the following expression is valid:

where such that and . Moreover, from (4.4) and (2.17), one has

Substituting (4.7) into (4.6), we have

where . It follows from the condition (, ) for that there exists a constant , such that and

So, the zero solution of (4.1) is asymptotically stable.

### 4.2. The Caputo Derivative Case

In this subsection, we consider a nonautonomous fractional differential system with Caputo derivative

under the initial conditions

where , , and are as in Section 4.1, is a continuously differentiable matrix. We can get the solution of (4.10)-(4.11) by using the Laplace transform and Laplace inverse transform

The main stability results of this subsection are derived as follows.

Theorem 4.3.

If the matrix such that , , the critical eigenvalues which satisfy have the same algebraic and geometric multiplicities, and is bounded, then the zero solution of (4.10) is stable.

Proof.

The proof line is similar to that of Theorem 4.1.

Theorem 4.4.

If the matrix such that , , and (, ) for , then the zero solution of (4.10) is asymptotically stable.

Proof.

From the solution (4.12) and Lemma 2.7, we can directly get

where and , such that . Furthermore, there exists a constant such that

that is,

Substituting (4.15) into (4.13) gives

It follows from the condition (, ) for that there exists a constant , such that and

So, the zero solution of (4.10) is asymptotically stable.